Where did I write that scientific belief is infallible? And who is this "we" to which you refer? Are you a room full of Zs and zs, all watching the screen simultaneously? Or, perhaps one physical Z with many disassociated identities? I don't "believe" that science is infallible. Come to think of it, I don't "believe" in anything. I just weigh the available evidence and try to reach a reasoned conclusion. And, the available evidence in this case, weighs in favor of the chicken preceding the egg. If you have some contrary evidence, I'm sure that "we" would all enjoy reading it.
What a chicken is, is subjective and entirely defined by man. There are no strict genetic rules or definite boundaries on what constitutes a chicken. Distant ancestors of modern chickens will look different, and even further back they will look so different that many people will disagree they can be called chickens at all. Equally take poodles - to get a poodle nowadays you mate two existing poodles together. You don't see wolves, or other dog breeds mating together to produce poodles. Yet poodles did once originate from other dog breeds. The trick is that it didn't happen in one generation - but over many many generations so that certain traits could gradually fix. Further back in the poodle ancestory the old poodles will look less similar to poodles today and more like the breeds poodles originated from. There will be a point back in time where many people will disagree they are poodles at all. So yes the egg did come before the chicken, just as the dog embryo came before the poodle.
Regardless, at one point in the past, the last change in an allele of the genetics of the modern chicken occurred, from some creature that may have looked indistinguishable from a modern chicken, but was not. That creature, may have even laid eggs, and the allele change may have even occurred while the chicken was inside the egg. But, until that creature cracked its shell, grew up and laid an egg containing a modern chicken, there could not be a modern chicken egg. Therefore, the chicken preceded the egg. Of course, I could be wrong: http://www.cnn.com/2006/TECH/science/05/26/chicken.egg/
Now that I think about it I think the whole idea of a point at which non-chicken becomes a chicken is the problem. Afterall humans invented the group called "chicken", and we have defined that group so vaguely that the edges gradually fade out. So there is no clear boundary to expect between chicken and non chicken. It's not that there is a specific boundary out there but we haven't the tools or knowledge to find it yet. It's literally that there is no specific boundary. Depends on whether the question is "what came first the chicken or the generic egg" or "what came first the chicken or the chicken egg" Two similar questions with very different answers.
After thinking about this more I have decided that the question is at fault as it assumes a definable point at which a non-chicken became a chicken. It's like the equally flawed question "what race was the mother who gave birth to the first asian?"